Problem: The fraction $p$ of the population who has heard a breaking news story increases at a rate proportional to the fraction of the population who has not yet heard the news story. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{dp}{dt}=kp$ (Choice B) B $\dfrac{dp}{dt}=k(1-p)$ (Choice C) C $\dfrac{dp}{dt}=-kp$ (Choice D) D $\dfrac{dp}{dt}=1-kp$
Solution: The fraction of the population who has heard the news story is denoted by $p$. The rate of change of that fraction is represented by $p'(t)$, or $\dfrac{dp}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the fraction of the population who has not yet heard the news story, $1-p$. In conclusion, the equation that describes this relationship is $\dfrac{dp}{dt}=k(1-p)$.